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Reversible Conservative Rational Abstract Geometrical Computation Is Turing-Universal

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Logical Approaches to Computational Barriers (CiE 2006)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 3988))

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Abstract

In Abstract geometrical computation for black hole computation (MCU ’04, LNCS 3354), the author provides a setting based on rational numbers, abstract geometrical computation, with super-Turing capability. In the present paper, we prove the Turing computing capability of reversible conservative abstract geometrical computation. Reversibility allows backtracking as well as saving energy; it corresponds here to the local reversibility of collisions. Conservativeness corresponds to the preservation of another energy measure ensuring that the number of signals remains bounded. We first consider 2-counter automata enhanced with a stack to keep track of the computation. Then we built a simulation by reversible conservative rational signal machines.

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References

  1. Adamatzky, A. (ed.): Collision based computing. Springer, Heidelberg (2002)

    MATH  Google Scholar 

  2. Asarin, E., Maler, O.: Achilles and the Tortoise climbing up the arithmetical hierarchy. In: Thiagarajan, P.S. (ed.) FSTTCS 1995. LNCS, vol. 1026, pp. 471–483. Springer, Heidelberg (1995)

    Chapter  Google Scholar 

  3. Bennett, C.H.: Logical reversibility of computation. IBM J. Res. Dev. 6, 525–532 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bennett, C.H.: Notes on the history of reversible computation. IBM J. Res. Dev. 32(1), 16–23 (1988)

    Article  MathSciNet  Google Scholar 

  5. Boccara, N., Nasser, J., Roger, M.: Particle-like structures and interactions in spatio-temporal patterns generated by one-dimensional deterministic cellular automaton rules. Phys. Rev. A 44(2), 866–875 (1991)

    Article  Google Scholar 

  6. Bournez, O.: Achilles and the Tortoise climbing up the hyper-arithmetical hierarchy. Theoret. Comp. Sci. 210(1), 21–71 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  7. Čulik II, K.: On invertible cellular automata. Complex Systems 1, 1035–1044 (1987)

    MathSciNet  MATH  Google Scholar 

  8. Durand-Lose, J.: Reversible cellular automaton able to simulate any other reversible one using partitioning automata. In: Baeza-Yates, R., Poblete, P.V., Goles, E. (eds.) LATIN 1995. LNCS, vol. 911, pp. 230–244. Springer, Heidelberg (1995)

    Chapter  Google Scholar 

  9. Durand-Lose, J.: Intrinsic universality of a 1-dimensional reversible cellular automaton. In: Reischuk, R., Morvan, M. (eds.) STACS 1997. LNCS, vol. 1200, pp. 439–450. Springer, Heidelberg (1997)

    Chapter  Google Scholar 

  10. Durand-Lose, J.: Computing inside the billiard ball model. In: Adamatzky, A. (ed.) Collision-based computing, pp. 135–160. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  11. Durand-Lose, J.: Abstract geometrical computation 1: embedding black hole computations with rational numbers. Research Report RR-2005-05, LIFO, U. D’Orléans, France (2005), http://www.univ-orleans.fr/lifo/prodsci/rapports

  12. Durand-Lose, J.: Abstract geometrical computation for black hole computation. In: Margenstern, M. (ed.) MCU 2004. LNCS, vol. 3354, pp. 176–187. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  13. Durand-Lose, J.: Abstract geometrical computation: Turing-computing ability and undecidability. In: Cooper, S.B., Löwe, B., Torenvliet, L. (eds.) CiE 2005. LNCS, vol. 3526, pp. 106–116. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  14. Delorme, M., Mazoyer, J.: Signals on cellular automata. In: [Ada02], pp. 234–275 (2002)

    Google Scholar 

  15. Dubacq, J.-C.: How to simulate Turing machines by invertible 1d cellular automata. International Journal of Foundations of Computer Science 6(4), 395–402 (1995)

    Article  MATH  Google Scholar 

  16. Fredkin, E., Toffoli, T.: Conservative logic. International Journal of Theoretical Physics 21(3/4), 219–253 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  17. Ilachinski, A.: Cellular automata –a discrete universe. World Scientific, Singapore (2001)

    Book  MATH  Google Scholar 

  18. Jacopini, G., Sontacchi, G.: Reversible parallel computation: an evolving space-model. Theoret. Comp. Sci. 73(1), 1–46 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  19. Jakubowsky, M.H., Steiglitz, K., Squier, R.: Computing with solitons: a review and prospectus. In: [Ada 2002], pp. 277–297 (2002)

    Google Scholar 

  20. Kari, J.: Reversibility of 2D cellular automata is undecidable. Phys. D 45, 379–385 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kari, J.: Reversibility and surjectivity problems of cellular automata. J. Comput. System Sci. 48(1), 149–182 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kari, J.: Representation of reversible cellular automata with block permutations. Math. System Theory 29, 47–61 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  23. Lecerf, Y.: Machines de Turing réversibles. Récursive insolubilité en n ∈ N de l’équation u = θ n u, où θ est un isomorphisme de codes. Comptes rendus des séances de l’académie des sciences 257, 2597–2600 (1963)

    MathSciNet  Google Scholar 

  24. Lindgren, K., Nordahl, M.G.: Universal computation in simple one-dimensional cellular automata. Complex Systems 4, 299–318 (1990)

    MathSciNet  MATH  Google Scholar 

  25. Li, M., Tromp, J., Vitanyi, P.: Reversible simulation of irreversible computation. Physica. D 120, 168–176 (1998)

    Article  Google Scholar 

  26. Minsky, M.: Finite and infinite machines. Prentice Hall, Englewood Cliffs (1967)

    MATH  Google Scholar 

  27. Morita, K.: Computation-universality of one-dimensional one-way reversible cellular automata. Inform. Process. Lett. 42, 325–329 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  28. Morita, K.: Reversible simulation of one-dimensional irreversible cellular automata. Theoret. Comp. Sci. 148, 157–163 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  29. Morita, K.: Universality of a reversible two-counter machine. Theoret. Comp. Sci. 168(2), 303–320 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  30. Mazoyer, J., Terrier, V.: Signals in one-dimensional cellular automata. Theoret. Comp. Sci. 217(1), 53–80 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  31. Toffoli, T., Margolus, N.: Invertible cellular automata: a review. Phys. D 45, 229–253 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  32. Toffoli, T.: Computation and construction universality of reversible cellular automata. J. Comput. System Sci. 15, 213–231 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  33. Toffoli, T.: Reversible computing. In: de Bakker, J.W., van Leeuwen, J. (eds.) ICALP 1980. LNCS, vol. 85, pp. 632–644. Springer, Heidelberg (1980)

    Chapter  Google Scholar 

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Author information

Authors and Affiliations

  1. Laboratoire d’Informatique Fondamentale d’Orléans, Université d’Orléans, B.P. 6759, F-45067 Cedex 2, ORLÉANS

    Jérôme Durand-Lose

Authors
  1. Jérôme Durand-Lose
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Editor information

Editors and Affiliations

  1. Department of Computer Science, Swansea University, Singleton Park, SA2 8PP, Swansea, United Kingdom

    Arnold Beckmann

  2. Department of Computer Science, University of Wales Swansea, Singleton Park, SA2 8PP, Swansea, UK

    Ulrich Berger

  3. Universiteit van Amsterdam, Plantage Muidergracht 24, 1018, Amsterdam, TV, The Netherlands

    Benedikt Löwe

  4. School of Physical Sciences, Swansea University, Singleton Park, SA2 8PP, Swansea, Wales, United Kingdom

    John V. Tucker

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© 2006 Springer-Verlag Berlin Heidelberg

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Durand-Lose, J. (2006). Reversible Conservative Rational Abstract Geometrical Computation Is Turing-Universal. In: Beckmann, A., Berger, U., Löwe, B., Tucker, J.V. (eds) Logical Approaches to Computational Barriers. CiE 2006. Lecture Notes in Computer Science, vol 3988. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11780342_18

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  • DOI: https://doi.org/10.1007/11780342_18

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-35466-6

  • Online ISBN: 978-3-540-35468-0

  • eBook Packages: Computer ScienceComputer Science (R0)

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